The science of numbers otherwise known as mathematics is various disciplines and principles that involve changes in structure, space, quantity and change. It has been a fundamental of many other sciences that rely on numbers for reasoning. Being said that, mathematics need to have learning standards. Because having standards in mathematical learning will give students realistic and clear expectations of what they are about to learn in math disciplines. Students will also have understanding of the reasons why learning math is detrimental in connecting concepts to their previous learning and how they apply in practical situations.

How do standards improve mathematics instruction? A carefully structured curriculum allows more effective approach in teaching mathematics. Standards provide clear guideline on how the broad topics should be broken down into appropriate learning level. It entails student learning achievement by gaining clarity and increases their common understanding of the complex instruction to a simpler point perspective. The process of breaking down the broad concepts allows teachers to develop their own interpretation on the topic, thus allowing a more personal approach in teaching style that the students can easily comprehend. Improvement process also delivers stronger content knowledge for teachers wherein the interpretation of mathematical instructions are not only based on their own teaching perspectives and experience but from within the acceptable standard parameters (Calstate.edu. 2009).

A consistent and very predictable approach in teaching math is being practiced in many classrooms not only in the United States but also in some parts of the world. The approach in which memorization of information along with developing ability to follow the rules and execute procedure and apply formulas exist in traditional mathematics which only students who has the capability to absorbing, regurgitating and accumulate information in this approach can actually excel. In contradiction to traditional mathematics or otherwise behaviorist approach, constructivist or reform-oriented math focuses on the student’s new tasks based on prior knowledge. It assimilates new initiatives in information feeding and develops easier to understand meanings (Nesmith, Suzanne J. N.D.).

The limitations of traditional mathematics program have something to do with how the teachers were oriented for instructions. Mathematical knowledge is ready-made before being presented to the students who in return are not receptive yet to the ideas. The problem with this approach is that the students are exposed to the procedures without their full understanding. When mathematical procedures are presented first before the students had the chance to absorb the concepts and the meaning of the topic, it is likely that they thinking ability would be overwhelmed by the instructions (Roh, Kyeong Ha 2003). For constructivist type learning the limitations are set by the teacher’s unique personal experience it is because knowledge is totally subjective. By interjecting a constructive approach in teaching, students would have a prior idea of the subject and what seems to limit the educator is his own personal experience of delivering the lessons.

In the sample lesson on Continuous Distribution, the objectives of the lesson is to determine the continuous and discrete random variables, stating the parameters and the use of normal, exponential and uniform values, selecting the context problem in continuous distribution appropriately, sketching normal, exponential and uniform probability in density areas and functions, solving the X of a given area from the exponential, normal and uniform model. Other objectives include knowing the triangular distribution being used in theory analysis.

The lesson objectives addressed the standards in learning focusing on content areas of numbers and operation, algebra and geometry. This also includes the process of problem solving and representation. The importance of following the standards in this area is to identify the link between algebra and geometry because sometimes individuals see mathematics as facts and procedures, and the students should realize the connections between these two stands of mathematics to reinforce the concept development as basis for more advanced mathematical applications (Nctm.org N.D.).

In the given sample topic lesson of Continuous Distribution, the methods used to represent the lesson is by giving mathematical problems to solve using graphs and equation and as an example to be able to describe continuous distribution a line graph was presented for both CDF (cumulative distribution function) and PDF (probability density function). By plotting data in a X and Y axis, the difference in variables were easily pointed out because of the difference on how the curves appear on the graph. Based on the lesson plan, it seems that the instructions were presented uniformly in all classrooms, because aside from the fact that the lesson plan was already been dissected to different topics therefore the teacher will just go with the flow of the lesson plan without any regards to classroom diversity. Being said that it shows that there was no difference on how the lessons were delivered in relation to classroom diversity.

In term of using technological means to aide in presenting the lesson there were a coupe of essential technological tools necessary to effectively disseminate instructions. One is the use of scientific calculators; we all know how handy it is to have a calculator on hand but with the kind of mathematical stand and standard adhere upon on the lesson scientific calculator is one of the best tools to aide learning. Drawing the equation on the board is one thing but solving the give problem is another. The complexity of mathematical operation needed to obtain reasons for the mathematical problem requires a fast calculating tool. Scientific calculators are designed to incorporate preset numerical and equation functions which eliminates the need to do separate calculations on other mathematical symbols. Another effective tool used for his lesson is a computer, only a single computer is needed plus an overhead project is enough to demonstrate the graphical representation of the problem and to show the difference in the curve lines. With a help of a computer to demonstrate the graphs to show CDF and PDF differences it would have been more difficult to explain and manually plot the data to show results.

In this kind of lesson, there is no need for any manipulative objects to aide learning. Mathematical operations and instructions needed a close attention from the students and since is more of instructions, manipulative objects is not needed to effectively explain the process. Unlike subjects like science or anatomy, objects are needed to show characteristics of the topic being discussed but with math the use of objects would only cause distraction and confusion. There are several learning assessment techniques that can be used for mathematics lesson. One of the most effective is problem recognition and solving. Math involves a lot numbers to decipher and to assess understanding of the topic, problem solving exercises must be given in order to test student comprehension of the lesson. Muddiest point technique is similar to short quiz or minute paper but the difference is that the student must answer the lesson is all about and ask how much they have understand the lecture, this would entail questions that will reveal areas in the lecture that they do not have a full understanding of. The next one is background knowledge check; this will allow the teacher o test prior knowledge of the students and to gauge familiarity to the subject. This will also determine what lessons needs to be recalled and it would make the teacher’s work easier rather than spending time discussing lesson topic that the students are already aware about (Cmu.edu N.D.).

If were given a chance to change any of the lessons, I would suggest to use more practical examples in the lesson plan because discussing numbers and more words eradicates interests in many students. The first five minutes of classroom lessons determines the outcome of the entire lesson discussion. It is crucial that the student understands the course of the lesson at the very beginning of the session. Stair step approach should have been observed wherein the lesson will start from a vey simple problem that demands very simple solution using the core principles of the topic. Sudden jump to complex studies creates shock to the students and the tendency is they would loose interest to learn because of their perception that the lesson is impossible to understand. Testing the water first would be a lot easier for the educator in any subject to disseminate learning and to se fist which areas needs to be recalled or if the students needs a more advanced course of discussion.

Once more thing that should be practiced in much mathematic class is the use of layman terminologies, because the mathematical operations and instructions are already quite challenging and then adding it with terminologies that although correct are still shocking strike to the student’s thinking machines. Avoid jargons as much as possible; explain terms and instructions in the manner that students can easily comprehend. Sometimes taking advantage of the student’s common interest will work wonders in feeding them the lesson. Pop culture is very appealing to students, why not use it to make them more interested on learning and initiate easier explanations by means using songs or make a rap song out of the given mathematical problem and the same thing goes for the instructions. Since students can memorize songs easily, I would be bet to utilize that opportunity to make them memorize the lesson and instructions easily as well, by doing so would also create a lively environment during the class and make them learn mathematics effectively.

## References

Calstate.edu (2009) IMPROVING STANDARDS-BASED INSTRUCTION Web Retrieved on February 12, 2012 from http://www.calstate.edu/capp/projects/docs/Improving_Standrs-Base_Instruct-acc.pdf

Nesmith, Suzanne J. (N.D.) Mathematics and Literature: Educators’ Perspectives on Utilizing a Reformative Approach to Bridge Two Cultures Web Retrieved on February 12, 2012 from http://forumonpublicpolicy.com/summer08papers/archivesummer08/nesmith.pdf

Roh, Kyeong Ha (2003) Problem-Based Learning in Mathematics ERIC Digest Web Retrieved on February 12, 2012 from http://www.ericdigests.org/2004-3/math.html

Nctm.org (N.D.) Guiding Principles for Mathematics Curriculum and Assessment Web Retrieved on February 12, 2012 from http://www.nctm.org/standards/content.aspx?id=23273

Cmu.edu (N.D.) Classroom Assessment Techniques Web Retrieved on February 12, 2012 from http://www.cmu.edu/teaching/resources/Teaching/CourseDesign/Assessment-Grading/Rubrics/ClassroomAssess.pdf